LETTERS PUBLISHED ONLINE: 29 JUNE 2014 | DOI: 10.1038/NNANO.2014.129

Quantum dots with single-atom precision Stefan Fo¨lsch1 *, Jesu´s Martı´nez-Blanco1, Jianshu Yang1, Kiyoshi Kanisawa2 and Steven C. Erwin3 * Quantum dots are often called artificial atoms because, like real atoms, they confine electrons to quantized states with discrete energies. However, although real atoms are identical, most quantum dots comprise hundreds or thousands of atoms, with inevitable variations in size and shape and, consequently, unavoidable variability in their wavefunctions and energies. Electrostatic gates can be used to mitigate these variations by adjusting the electron energy levels1–3, but the more ambitious goal of creating quantum dots with intrinsically digital fidelity by eliminating statistical variations in their size, shape and arrangement remains elusive4–9. We used a scanning tunnelling microscope to create quantum dots with identical, deterministic sizes. By using the lattice of a reconstructed semiconductor surface to fix the position of each atom, we controlled the shape and location of the dots with effectively zero error. This allowed us to construct quantum dot molecules whose coupling has no intrinsic variation but could nonetheless be tuned with arbitrary precision over a wide range. Digital fidelity opens the door to quantum dot architectures free of intrinsic broadening—an important goal for technologies from nanophotonics10 to quantum information processing11,12 as well as for fundamental studies of confined electrons13–17. Creating atomically precise quantum dots requires every atom of the quantum dot to be placed in a precisely specified location without error, and multiple dots to be arranged in exactly defined configurations without variation. We used a scanning tunnelling microscope (STM) to manipulate the atoms18 and a surface template to define a lattice of allowed atomic positions. The template was the InAs(111)A surface, which has a 2 × 2 In-vacancy reconstruction19 and a low concentration (roughly 0.005 monolayer) of native In adatoms adsorbed above the vacancy sites20. The adatoms are ionized þ1 donors20,21 and can be moved with the STM tip by vertical atom manipulation18. We assembled linear chains22, as illustrated by the example of an In22 chain in Fig. 1a. The adatoms reside √on  nearest-neighbour vacancy sites separated by a′ = a0 2 = 8.57 Å (where a0 ¼ 6.06 Å is the InAs lattice constant), as shown in Fig. 1b. A chain of ionized In adatoms forms a quantum dot by creating an electrostatic potential well that confines the electrons belonging to surface states of pristine InAs(111)A. The resulting quantumconfined states can be probed by scanning tunnelling spectroscopy (STS) measurements of the differential conductance. Conductance spectra of the In22 chain (Fig. 1c) show a series of resonances in the conduction band, labelled by the principal quantum number n. The corresponding experimental density-of-states (DOS) spatial maps in Fig. 1d reveal the wavefunctions of these quantized states, showing n lobes and n 2 1 nodes along the chain, as expected for a quantum particle in a box. The same qualitative behaviour is observed for a wide range of chain lengths, from N ¼ 6 to 25 (Supplementary Fig. 1). The energies of these states scale approximately as En ≈ n 2/N 2, as expected for confined electrons23. For

short chains (N ≤ 6) we find that the lowest-lying (n ¼ 1) state first emerges at 0.1 eV below the Fermi level (Supplementary Fig. 2). Despite their small size, these quantum dots do not exhibit Coulomb blockade because the mean carrier lifetime on the dot (estimated from the level broadening of the quantum states) is much shorter than the inverse tunnelling rate through it. Figure 2 traces, theoretically, the quantization of the intrinsic InAs surface states. We first used density-functional theory (DFT) with a screened hybrid functional24,25 to determine the electronic structure of pristine InAs(111)A-(2 × 2). Two surface states are formed within the InAs conduction band (Fig. 2a). These states arise predominantly from the unoccupied 6s and 6p atomic orbitals of In surface atoms. In the presence of In adatoms, both states undergo confinement. Our experiments probed only the lowerlying state; indeed, the theoretical dispersion E(k) of this state is in excellent agreement (Fig. 2b) with the energy versus wavenumber (k ¼ 2p/l ) derived experimentally from conductance spectra (yielding En) and DOS spatial maps (yielding l/2) for chain lengths N ¼ 13–25 and states n ¼ 2–7. To confirm this we next used DFT within the generalized-gradient approximation26 to visualize (Fig. 2c,d) the first two quantized states c1,2(r) of a quantum dot consisting of N ¼ 6 adatoms. The nodal structure of these states agrees well with the spatial maps in Fig. 1d, and their underlying surface atomic orbital structure is readily evident from the local maxima at the In surface sites. We note that these confined semiconductor surface states are different from previously reported states arising from the In orbitals on the adatom chain itself22 (Supplementary Fig. 3). Because the In adatoms are strictly confined to the lattice of vacancy sites, every quantum dot with N adatoms is essentially identical, with no intrinsic variation in size, shape or position. This suggests that quantum dot molecules consisting of coupled adatom chains will reflect the same invariance. Figure 3 demonstrates the predictable, smoothly decreasing quantum coupling between two identical In6 quantum dots separated by increasing distance. For reference, Fig. 3a shows a topographic image of a single In6 chain (upper panel), together with a DOS bias map (centre panel) and a DOS spatial map (lower panel) of the c1(r) confined state located 114 meV below the Fermi level. Figure 3b shows the same information for a quantum dot molecule consisting of two identical In6 chains separated by a gap of two empty vacancy sites. The DOS maps reveal the formation of bonding (s) and antibonding (s*) states due to interdot coupling. As expected from the symmetry, the amplitude at the midpoint of the quantum dot molecular wavefunction is either finite (for s) or vanishing (for s*). For this 2-vacancy gap, the energy splitting Ds2s* is 70 meV. The coupling between two separated quantum dots persists at remarkably large separations. Measurements of the coupling versus separation provide an experimental probe of invariance in quantum dot molecules, as well a challenge to theory to predict this dependence. Figure 3c shows the splitting Ds2s* versus

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Paul-Drude-Institut fu¨r Festko¨rperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany, 2 NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa, 243-0198, Japan, 3 Center for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, USA. * e-mail: [email protected]; [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology

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Figure 1 | Quantized states of a digital quantum dot in which electrons are confined by a chain of ionized adatoms. a, Topographic STM image (0.1 nA, 20.3 V) of a chain of In adatoms assembled on InAs(111)A. Twenty-two In atoms were placed on adjacent In-vacancy sites of the (2 × 2)-reconstructed surface. b, Atomic structure of the image section indicated in a. The surface consists of In (green) and As (orange) atoms, and the chain is formed by In adatoms (black circles) adsorbed above vacancy sites. c, Differential conductance (dI/dV) spectra (red and blue) recorded at the off-chain tip positions indicated in a, revealing quantized electron states with quantum numbers n ¼ 1–7. The reference spectrum of pristine InAs(111)A (green) reveals that the Fermi level is pinned in the conduction band due to intrinsic electron accumulation at the surface27. d, Spatial DOS maps D(x,y) obtained by constant-height dI/dV scanning at the bias voltages corresponding to the resonances in c. Quantized states for n ¼ 1–6, each with n lobes and n 2 1 nodes, are clearly revealed.

separation for vacancy gap sizes from 1 to 7 and Fig. 3d summarizes the mean values of Ds2s* for many independent structures assembled at different surface locations over different experimental runs. The variance in Ds2s* at a given separation is extremely small, with a standard deviation of 1 mV, confirming the high degree of fidelity of these quantum dot molecules. Theoretically, the observed splittings are reproduced by a tight-binding (TB) Hamiltonian in which electrons hop with effective amplitude tij ¼ t0exp(2dij/d0) between pairs of In adatoms. This effective hopping can be derived by considering multiple nearest-neighbour hops through the electrostatic barrier created by the vacancy gap (see Supplementary Discussion). We obtained the parameters t0 ¼ 225 meV and d0 ¼ 5.25a ′ by fitting the experimental results in Fig. 3d. The coupling in quantum dot molecules can be tuned with arbitrary precision and over a wide range (factor of two in Ds2s*) by placing additional adatoms in the gap region to provide auxiliary hopping pathways. We created a collection of about 30 different tuning configurations of the In6 quantum dot molecule and demonstrated a tuning resolution of 1 mV (Supplementary Figs 4 and 5). The effective TB hopping parameters derived above accurately describe the entire range of these experimentally tuned splittings. The resulting picture is that the tuning arises from auxiliary hopping pathways created between the two quantum dots, affording great flexibility as well as arbitrarily fine resolution. 2

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Figure 2 | Theoretical description of pristine and quantized surface states. a, Electronic structure of reconstructed InAs(111)A-(2 × 2) calculated by DFT with the HSE screened hybrid functional. Surface states with predominantly In s and p atomic character are yellow and blue, respectively, and lie within the bulk conduction band. Projected bulk bands are grey. b, Expanded view of the region outlined in a, showing the state formed by 6s orbitals of In surface atoms. This state becomes quantized when an In adatom chain is present. Black points are experimental energies versus wave vector, obtained spectroscopically from the quantized states exemplified in Fig. 1d. c,d, Quantized electron states from DFT with n ¼ 1 and 2 of an In6 chain. Cross-sections taken 3 Å above the surface emphasize their surface In atomic character. Line scans are one-dimensional averages on a log scale spanning six orders of magnitude. Large and small black circles denote In adatoms and In surface atoms, respectively.

Quantum dots with digital fidelity also simplify the task of creating, protecting and controlling degenerate states in quantum dot molecules, an important prerequisite for many technologies. In quantum computing, for example, qubits with doubly degenerate ground states offer protection against environmental decoherence11,12. By combining the invariance of quantum dot molecules with the intrinsic symmetry of the InAs(111)A vacancy lattice, we created degenerate states that are surprisingly resistant to environmental perturbations by defects. Consider the quantum dot molecule consisting of three In6 chains in an arrangement with C3v symmetry (Fig. 4a). Group theory implies the existence of a doubly degenerate state s* with eigenstates |s1*l ¼ (þ2, 21, 21) and |s2*l ¼ (0, þ1, 21) in the basis of the three quantum dots. The DOS spatial maps in Fig. 4a are consistent with the nondegenerate ground state |sl ¼ (þ1, þ1, þ1) (centre) and the superposition of the degenerate excited states |s1*l and |s2*l (right). An additional ‘foreign’ atom deliberately placed x sites from the origin (Fig. 4b) indeed breaks the

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Figure 3 | Antibonding states of two weakly coupled adatom chains forming a digital quantum dot molecule. a, Topographic STM image (0.1 nA, 20.3 V) of a single In6 chain (upper panel), a DOS bias map D(x,V) versus bias voltage V and position x along dashed line in upper panel (centre panel), and a spatial DOS map of the confined ground state (lower panel). b, Topographic STM image (0.1 nA, 20.3 V) of two In6 chains separated by a gap of two vacancy sites. The DOS bias map (centre panel) reveals bonding (s) and antibonding (s*) states at 2167 meV and 297 meV, respectively. The symmetric (s) and antisymmetric (s*) character of the states is evident in their DOS spatial maps (lower panel). c, Conductance spectra quantifying the s2s* splitting Ds2s* of two In6 chains for different vacancy gap sizes. Spectra were recorded at the position marked in the top panel of b. d, Mean splittings Ds2s* obtained for two In6 chains versus vacancy gap size (empty circles). The standard deviation is 1 mV. Solid circles in d show theoretical s2s* splittings calculated within a TB model for hopping across the gap.

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Figure 4 | Protected degeneracy in a digital triple dot. a, Topographic STM image (left, 0.1 nA, 0.1 V) of three In6 chains in an arrangement with three-fold symmetry. DOS spatial maps reveal the ground state s (centre) and the doubly degenerate excited state s* (right). b, As in a, but with an additional ‘foreign’ In adatom at x ¼ 24. The two-fold degeneracy of the s* state is undisturbed to within experimental resolution. c, Conductance spectra recorded at the positions marked in a and b demonstrate that the ionized foreign atom does not detectably lift the s* state degeneracy. d, Topographic STM image of the triple dot (0.1 nA, 0.1 V) after its symmetry was deliberately broken by displacing the horizontal chain by one In vacancy spacing (Dx ¼ 1). The s* state splits into a s1*2s2* doublet given by eigenvectors |s1*l ¼ (þ2, 21, 21) and |s2*l ¼ (0, þ1, 21) in the basis of the quantum dots. e, Conductance spectra showing the s1*2s2* splitting for various Dx. f, Average s and s* energies (open circles) of the triple dot as a function of Dx. The splitting of the s* level establishes the two-fold degeneracy of this triple dot. Filled circles show energies calculated within the TB model. NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology

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degeneracy, but only very weakly; in our TB model the largest symmetry breaking (which occurs for x ¼ 24 in units of a ′ ) is only 2 meV (Supplementary Fig. 6). DOS spatial maps (Fig. 4b) and conductance spectra (Fig. 4c) confirm that the degeneracy is protected within our experimental resolution. To complete the demonstration of a degenerate s* state we next created a larger perturbation of the triple quantum dot molecule. In Fig. 4d we displaced the horizontal chain by one vacancy site (Dx ¼ 1). The resulting DOS spatial maps (Fig. 4e) reveal two distinct states, which we identify as s1* and s2*. The resulting s1*2s2* splitting is 16 mV and increases for larger displacements Dx ¼ 2 and 3 (Fig. 4e). These splittings are accurately reproduced by our TB model (Fig. 4f ) and confirm the two-fold degeneracy expected for the unperturbed triple quantum dot molecule. The fidelity offered by digital quantum dots also makes them excellent candidates for studying fundamental physics that is typically obscured by stochastic variations in size, shape or position of the chains. One example is the emergence of Peierls instabilities in one-dimensional metallic chains as their length is varied13. Others include the search for Majorana fermions in one-dimensional systems with appreciable spin–orbit coupling14,15, the demonstration of entangled states in quantum dot molecules16 and the investigation of the Kondo effect in quantum dots17. We anticipate that the elimination of uncontrolled variations in quantum dot architectures will offer benefits to a broad range of future technological applications in which digital fidelity is important.

Methods Sample preparation. The experiments were performed with an STM operated in ultrahigh vacuum (UHV) at a sample temperature of 5 K. Undoped InAs layers with thickness of 20 nm were grown by molecular beam epitaxy (MBE) on an InAs(111)A wafer (purchased from Wafer Technology) to prepare the In-terminated InAs(111)A-(2 × 2) surface. After the MBE growth, the surface was capped by an amorphous arsenic layer and transferred under ambient conditions to the UHVSTM system. The As capping layer was desorbed by annealing at 630 K in UHV before loading the sample into the STM. Tips were prepared by electrochemical etching of a polycrystalline tungsten wire and cleaned in UHV by electron-beam heating and Ne-ion sputtering. Spectroscopy measurements of the differential tunnelling conductance were performed by a lock-in technique (5–10 mV peak-topeak modulation at a frequency of 675 Hz) with disabled feedback loop. Bias voltages refer to the sample with respect to the STM tip. Native In adatoms on InAs(111)A were repositioned by vertical atom manipulation as described in ref. 22. DFT calculations. First-principles calculations were used to determine the equilibrium geometry and electronic band structure of the In-terminated InAs(111)A-(2 × 2) surface. Total energies and forces were calculated within the generalized-gradient approximation of Perdew, Burke and Ernzerhof (PBE)26 to DFT, using projectoraugmented-wave (PAW) potentials as implemented in VASP24. The electronic band structure was calculated within the screened hybrid functional of Heyd, Scuseria and Ernzerhof (HSE)25. The plane-wave cutoff for all calculations was 250 eV. Within DFT/HSE, the predicted lattice constant (6.11 Å) and bandgap (0.42 eV) for bulk InAs are extremely close to their experimental values (6.06 Å and 0.42 eV). The surface calculations were performed in a slab geometry with eight layers of InAs and a vacuum region of 10 Å. All atomic positions were relaxed, except the bottom InAs layers and the passivating pseudohydrogen layer, until the largest force component on every atom was below 0.02 eV Å21. The sampling of the 2 × 2 surface Brillouin zone was carried out with a 5 × 5 Monkhorst–Pack mesh centred at the G point. The equilibrium position of individual In adatoms adsorbed on the InAs(111)A(2 × 2) surface was determined within DFT/PBE (HSE is intractable for large systems). At equilibrium the adatoms are 1.7 Å above the InAs surface atomic plane and are centred above the vacancy site of a missing surface In atom. Chains of In adatoms were constructed by repeating this locally relaxed adatom configuration at adjacent surface vacancy sites. Additional relaxation within a chain was found to be negligible. A large surface supercell was used to calculate the n ¼ 1 and 2 quantum-confined states for the N ¼ 6 adatom chain in Fig. 2c,d. This surface supercell was rectangular with dimensions of 104 Å × 45 Å and contained a total of 1,086 In and As atoms in four InAs layers.

Received 18 February 2014; accepted 27 May 2014; published online 29 June 2014

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Acknowledgements The authors thank C.S. Hellberg, Al. L. Efros and E.I. Rashba for discussions. This work was supported by the German Research Foundation (FO 362/4-1) and the Office of Naval Research through the Naval Research Laboratory’s Basic Research Program. Computations were performed at the DoD Major Shared Resource Center at AFRL.

Author contributions J.Y., J.M-B., and S.F. performed the experiments. S.F. carried out the experimental data analysis. K.K. performed the MBE growth of the samples and participated in discussions of the results. S.C.E. performed the DFT calculations and tight-binding theoretical modelling. S.F. and S.C.E. wrote the manuscript.

Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to S.F. and S.C.E.

References 1. Klein, D. L., Roth, R., Lim, A. K. L., Alivisatos, A. P. & McKuen, P. L. A singleelectron transistor made from a cadmium selenide nanocrystal. Nature 389, 699–701 (1997). 4

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Competing financial interests The authors declare no competing financial interests.

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